first of all I'd like to clarify that my biggest problem with this topic is probably my inability to formulate it correctly, maybe the answer is found trivially on the internet but I'm not able to find it. That being said I'll ask the question.
Let's say I am given some kind of probability sequence that correlates a series of independant events with only two possible outcomes. For example: I am given the probability of rain (either it rains or it doesn't) for every day for a year. I would like to somehow cuantify how good this prediction is when the only thing I can do is to check where it rained or not those days.
My first aproach is that, if I had large ammounts of data or the probabilities aren't too different fron eachother what I could do is divide the set of probabilities in subsets "bins" like the probabilities from 20-25% where I have let's say 100 events, then I could do some kind of usual statistical analysis, asuming normal distribution etc. If this is the correct aproach I would like to know how this is called so I can do more research and also would like to know how to calculate the size of those bins etc. Maybe what to do if you have different/same unceirtanty of each of the probabilities given.
If this is not the correct aproach I'd appreaciate some guidelines on how to go on. Bottom line, I want to know that if the n+1 term of the succession says the event has a probability p with some unceirtanty given (or not) What is a reasonable way to check this based on previous data and how to recalculate the probability with it's new unceirtanty (bigger I presumme).
Thanks a lot and sorry for the lack of mathematical rigor in the question (I would have probably obscured it really) and also sorry for the poor writting, as english is not my main languague, but I think I have the sufficient mathematical background to understand the answer, the thing I want the most is a starting point (name of method etc.) so I can research on my own.